Intro to Probability

Applications

Scientists use something called 5-sigma statistical significance to confirm phenomena observed at the Large Hadron Collider, meaning that such an event is 5 standard deviations or more away from what one would expect if it happened by chance. This translates to a probability of approximately \(3\times 10^{-7}.\)

Recently, scientists at CERN have attributed a recent observation of a new particle to "statistical fluctuation." The significance of the observation was downgraded to 2-sigma, which translates to a probability of approximately \(0.05\) that the observation would have happened by chance.

Approximately how many times more likely would an observation of 2-sigma significance be than an observation of 5-sigma significance?

A weather forecast predicts a 60% chance of rain in a specific area for a certain day. It does not rain in that area during that day. Which of these conclusions about the forecast is the most reasonable?

A: With a 60% chance of rain, it should have at least rained a little bit.

B: The weather forecast was flawed, and needs to be corrected for the future.

C: It cannot be concluded from a single day of data whether the forecast is accurate or not.

In WWII, enemies would engage in plane-to-plane aerial combat. Unsurprisingly, many planes were lost to crashes; if a bullet strikes a plane in a sensitive area, it's very hard to make it back to base. For the planes that did come back, the mechanics kept track of the location of bullet holes in the fuselage, so that they could reinforce the planes in the most vulnerable locations.

For American planes, the bullet holes on returning planes were distributed as follows:

Where should the mechanics reinforce planes so that more of them come back safely?

Which of the following statements is the best comparison between the probabilities used in weather forecasting and the probabilities used to describe dice rolls and coin flips?

A: Weather probabilities are completely subjective measures of likelihood, while the probabilities of dice rolls and coin flips are completely objective measurements of likelihood.

B: Weather probabilities, like the probabilities of dice rolls and coin flips, are based on objective methods and measurements. However, the probabilities of dice rolls and coin flips can be tested over many trials of the same controlled experiment, while weather probabilities cannot be tested in the same way.

C: The outcome of future weather behaves just like the outcome of the roll of a die. There are a set of weather outcomes that are each equally likely, and one of those outcomes is chosen at random.